1. ## [SOLVED] Related Rate

Helium is pumped into a spherical balloon at a rate of 3 cubic feet per second. How fast is the radius increasing after 3 minutes?

I know . but i don't know how to find the radius of the balloon at 3 minutes.

2. Note: Your hotlinked image from UGA is generating certificate errors. You might want to replace the image with the math tools provided here: $V\, =\, \frac{4}{3}\pi r^3$

Originally Posted by yeloc
Helium is pumped into a spherical balloon at a rate of 3 cubic feet per second. How fast is the radius increasing after 3 minutes?
You are given that $\frac{dV}{dt}\, =\, 3$. You know the "volume" formula.

Convert the "three minutes" into seconds, and find the volume after that number of seconds.

Plug this volume into the "volume" formula, and solve for the radius at that time.

Differentiate the "volume" formula with respect to time. Plug in the known values for $\frac{dV}{dt}$ and $r$, and solve for $\frac{dr}{dt}$.

3. Originally Posted by yeloc
Helium is pumped into a spherical balloon at a rate of 3 cubic feet per second. How fast is the radius increasing after 3 minutes?

I know . but i don't know how to find the radius of the balloon at 3 minutes.
1. The volume of a sphere is calculated by:

$V=\dfrac43 \pi r^3~\implies~r=\sqrt[3]{\dfrac{3V}{4\pi}}$

2. The volume after 3 min = 180 s is 540 cft.

3. Differentiate the equation for r:

$\dfrac{dr}{dt} = \dfrac{d\left( \sqrt[3]{\dfrac{3V}{4\pi}} \right)}{dV} \cdot \dfrac{dV}{dt}$

4. Plug in V = 540 and $\dfrac{dV}{dt} = 3$ to calculate the rate of change of the radius.
Spoiler:
I've got $0.0094\ \dfrac{ft}s$

4. that answer is incorrect. I'm so confused!

5. Originally Posted by yeloc
that answer is incorrect. I'm so confused!
What result do you have?

6. I also got .009 according to your work, but it isn't the correct answer. and I don't know where the mistake is.

7. Originally Posted by yeloc
I also got .009 according to your work, but it isn't the correct answer. and I don't know where the mistake is.
Nor will we "know where the mistake is", until you've shown your work and reasoning.

8. I figured out how to work it out. The answer that it accepted was 0.00935

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### helium is pumped into a spherical balloon at a rate of 3

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